Least Squares.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Matplotlib created a temporary config/cache directory at /tmp/matplotlib-b8htc_z_ because the default path (/tmp/cache/matplotlib) is not a writable directory; it is highly recommended to set the MPLCONFIGDIR environment variable to a writable directory, in particular to speed up the import of Matplotlib and to better support multiprocessing.\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import numpy.linalg as la\n",
    "import scipy.linalg as sla\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lesson 9: Linear Least Squares"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Least Squares and Normal Equations\n",
    "\n",
    "Consider a set of $m$ data points $\\{(t_1,y_1),(t_2,y_2), ...,(t_m,y_m)\\}$. Suppose we want to find a straight line that best fits these data points. Mathematically, we want to find the slope and intercept ($x_1$ and $x_2$ respectively) such that\n",
    "\n",
    "$$y_i = x_1 t_i + x_2, \\qquad i \\in \\{1,2,...,m\\} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can write this more explicitly as ${\\bf y} = {\\bf t}\\cdot x_1 + 1\\cdot x_2$, which lets us separate the coefficients we are trying to solve for ($x_1$ and $x_2$) from the input data that we have (${\\bf y}$ and ${\\bf t}$).  Now we can put these values into a matrix format:\n",
    "\n",
    "$$\\begin{bmatrix}y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_m\\end{bmatrix} = \\begin{bmatrix}t_1 & 1 \\\\ t_2 & 1 \\\\ \\vdots & \\vdots \\\\ t_m & 1\\end{bmatrix} \\begin{bmatrix}x_1 \\\\ x_2\\end{bmatrix} \\qquad \\Leftrightarrow \\qquad {\\bf y} = {\\bf A}{\\bf x}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where ${\\bf A}$ is the design matrix that is a function of the $t_i$ data, ${\\bf y}$ is the vector with the $y_i$ data and ${\\bf x}$ is the vector with the coefficients $x_i$.  Notice how the values that multiply our coefficients $x_i$ are the columns of ${\\bf A}$.\n",
    "\n",
    "Generally, if we have a linear system where ${\\bf A}$ is an $m \\times n$ matrix and $m > n$ we call this system **overdetermined**. For these systems, the equality is usually not exactly satisfiable as ${\\bf y}$ may not lie in the column space of ${\\bf A}$. Therefore, an overdetermined system is better written as\n",
    "\n",
    "$${\\bf A}{\\bf x} \\cong {\\bf y}  $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For an overdetermined system ${\\bf A}{\\bf x} \\cong {\\bf y}$, we are typically looking for a solution ${\\bf x}$  that minimizes the squared norm of the residual vector ${\\bf r} = {\\bf y}-{\\bf A x}$,\n",
    "\n",
    "$$ \\min_{\\bf x} ||{\\bf r}||^2 = \\min_{\\bf x} ||{\\bf y} - {\\bf A x}||^2$$\n",
    "\n",
    "This problem  is called a **linear least-squares problem**, and the solution ${\\bf x}$ is called least-squares solution. You learned during lecture that the solution of the least squares problem is also the solution of the system of **normal equations**:\n",
    "\n",
    "$${\\bf A}^T {\\bf A}{\\bf x} = {\\bf A}^T {\\bf y}  $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 1 - fitting a line\n",
    "\n",
    "Consider the data set given by `pts`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2 3 5 6]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pts = np.array([[2, 4], [3, 2], [5, 1], [6, 0]])\n",
    "t_line = pts[:,0]\n",
    "\n",
    "\n",
    "y_line = pts[:,1]\n",
    "\n",
    "plt.plot(t_line, y_line, 'o')\n",
    "print(t_line)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Check your answers:**\n",
    "\n",
    "Define the matrix ${\\bf A}$ that can be used to solve for the coefficients $x_1$ and $x_2$.\n",
    "\n",
    "Store the value of it in variable `A_line`.\n",
    "\n",
    "*Hint: Try to define the design matrix programmatically (i.e. not hard-coded). Rather than iterating through a loop, you can directly construct ${\\bf A}^T$ as a NumPy array whose first entry is `t_line` and whose second entry is a vector of `1`s, then apply the transpose with `.T` to obtain ${\\bf A}$ as desired.*\n",
    "\n",
    "*Hint: To obtain the vector of `1`s with appropriate length, you may use `t_line**0` or `np.ones_like(t_line)`. Recalling the syntax for power in Python and broadcasting, can you reason why the first option works?*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[2. 1.]\n",
      " [3. 1.]\n",
      " [5. 1.]\n",
      " [6. 1.]]\n",
      "[[2 4]\n",
      " [3 2]\n",
      " [5 1]\n",
      " [6 0]]\n"
     ]
    }
   ],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "# Define A_line here\n",
    "A_line=np.ones((4,2))\n",
    "A_line[:,0] = pts[:,0]\n",
    "print(A_line)\n",
    "print(pts)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Check your answers:**\n",
    "\n",
    "Use the normal equations to find the coefficients in ${\\bf x}$.\n",
    "\n",
    "Store the value of it in variable `x_line`.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.9   5.35]\n"
     ]
    }
   ],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "x_line=la.solve(A_line.T @ A_line, A_line.T @ pts[:,1])\n",
    "print(x_line)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the points and fit curve:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fc58ab3ddc0>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_line, y_line, 'o')\n",
    "plt.plot(t_line, t_line * x_line[0] + x_line[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear Least Squares and QR"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "While easy to understand and remember, using the normal equations can be prone to numerical instabilities (you will learn more about this if you take a Numerical Methods or Analysis class! Hint hint!)\n",
    "\n",
    "Instead of solving the normal equations using the design matrix ${\\bf A}$, we will instead use its QR decomposition\n",
    "\n",
    "$$ {\\bf A} = {\\bf QR} $$\n",
    "\n",
    "where ${\\bf Q}$ is an orthogonal matrix and ${\\bf R}$ is an upper triangular matrix. Hence the least squares solution ${\\bf x}$ can be obtained as:\n",
    "\n",
    "$$ {\\bf A}^T{\\bf Ax} = {\\bf A}^T{\\bf y} \\quad \\Leftrightarrow \\quad {\\bf Rx} = {\\bf Q}^T{\\bf y}$$\n",
    "\n",
    "**Check your answers:**\n",
    "\n",
    "Write the `least_sq` function that uses QR decomposition and finds the least squares solution ${\\bf x}$ given matrix ${\\bf A}$ and vector ${\\bf y}$. Do not use the [numpy.linalg.inv](https://numpy.org/doc/stable/reference/generated/numpy.linalg.inv.html) function.\n",
    "\n",
    "*Hint: Try using the functions [numpy.linalg.qr](https://numpy.org/doc/stable/reference/generated/numpy.linalg.qr.html) and [scipy.linalg.solve_triangular](https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_triangular.html).*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "def least_sq(A, y):\n",
    "    # Find the least squares solution to Ax = y\n",
    "    # by solving Rx = Q^T y for x where A = QR\n",
    "    Q,R = la.qr(A)\n",
    "    x= sla.solve_triangular(R,Q.T@y)\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's confirm that we get the same answer as we did above with the normal equations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "xqr = least_sq(A_line, y_line)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fc58aab86a0>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_line, y_line, 'o')\n",
    "plt.plot(t_line, t_line * xqr[0] + xqr[1])\n",
    "plt.plot(t_line, t_line * x_line[0] + x_line[1],'x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 2: quadratic curve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets look at another set of points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.00000000e+00, 9.20033319e-01, 8.43398584e-01, 7.70095793e-01,\n",
       "       7.00124948e-01, 6.33486047e-01, 5.70179092e-01, 5.10204082e-01,\n",
       "       4.53561016e-01, 4.00249896e-01, 3.50270721e-01, 3.03623490e-01,\n",
       "       2.60308205e-01, 2.20324865e-01, 1.83673469e-01, 1.50354019e-01,\n",
       "       1.20366514e-01, 9.37109538e-02, 7.03873386e-02, 5.03956685e-02,\n",
       "       3.37359434e-02, 2.04081633e-02, 1.04123282e-02, 3.74843815e-03,\n",
       "       4.16493128e-04, 4.16493128e-04, 3.74843815e-03, 1.04123282e-02,\n",
       "       2.04081633e-02, 3.37359434e-02, 5.03956685e-02, 7.03873386e-02,\n",
       "       9.37109538e-02, 1.20366514e-01, 1.50354019e-01, 1.83673469e-01,\n",
       "       2.20324865e-01, 2.60308205e-01, 3.03623490e-01, 3.50270721e-01,\n",
       "       4.00249896e-01, 4.53561016e-01, 5.10204082e-01, 5.70179092e-01,\n",
       "       6.33486047e-01, 7.00124948e-01, 7.70095793e-01, 8.43398584e-01,\n",
       "       9.20033319e-01, 1.00000000e+00])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_quad = 50\n",
    "t_quad = np.linspace(-1, 1, n_quad)\n",
    "y_quad = t_quad ** 2 + np.random.randn(n_quad) * 0.05 + 5\n",
    "\n",
    "plt.plot(t_quad, y_quad, 'o')\n",
    "t_quad**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks a lot like a quadratic function!  Let's try to fit a curve to this:\n",
    "\n",
    "$$y = x_1 \\,t^2 + x_2\\,t + x_3$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Check your answer:**\n",
    "\n",
    "We want to find the coefficients $x_1$, $x_2$, and $x_3$ that solve the least squares problem ${\\bf A x} \\cong {\\bf y}$; first, we need to construct the design matrix. Taking ${\\bf x} = [x_1, x_2,x_3]^T$, define the design matrix ${\\bf A}$ corresponding to the model written above.\n",
    "\n",
    "Store the value of it in variable `A_quad`.\n",
    "\n",
    "*Hint: As with example A, try defining the matrix ${\\bf A}^T$ then apply the transpose to obtain ${\\bf A}$ in a single line of code. This time, what will the first entry (row of the transpose/column of the design matrix) be?*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "A_quad = np.array([t_quad**2,t_quad,np.ones_like(t_quad)]).T\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we solve for the coefficients ${\\bf x}$ using the `least_sq` function you have defined above. \n",
    "\n",
    "Store the value of it in variable `x_quad`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.98240922, 0.01898425, 4.99766062])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_quad = least_sq(A_quad, y_quad)\n",
    "x_quad"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6b24036310>]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_quad, y_quad, 'o')\n",
    "plt.plot(t_quad, x_quad[0] * t_quad ** 2 + x_quad[1] * t_quad + x_quad[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see, even though the resulting function does not appear linear, we are still able to use linear least squares here.  This is because the function only needs to be linear with respect to the coefficients $x_i$ (e.g. cannot make direct use of $x_i^2$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In fact, what we are doing here is simply taking linear combinations of some basis functions:\n",
    "\n",
    "$$ y = x_1 f_1(t) + x_2 f_2(t) + x_3 f_3(t) $$\n",
    "\n",
    "$$\\begin{align*} f_1(t) &= t^2 \\\\ f_2(t) &= t \\\\ f_3(t) &= 1 \\end{align*} $$\n",
    "\n",
    "As long as we can write the function we are trying to fit in this manner (or somehow get it in that form), we can use least squares."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example 3: exponential fit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assume the following set of data points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6b243b04f0>]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_exp = 50\n",
    "t_exp = np.linspace(0.1, np.e * 4, n_exp)\n",
    "y_exp = np.exp(t_exp * 0.5) * 3 + np.random.randn(n_exp) * 2\n",
    "y_exp = np.where(y_exp<1e-6, 1e-6, y_exp)\n",
    "plt.plot(t_exp, y_exp, 'o')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's say we want to fit the function\n",
    "\n",
    "$$ y = x_1\\,e^{x_2 \\, t} $$\n",
    "\n",
    "to the above plot. Notice that we can't directly use linear least squares, since the function is not linear with respect to every coefficient $x_i$.  What if we take the natural log of each side?\n",
    "\n",
    "$$ \\ln y = \\ln x_1 + x_2 t $$\n",
    "\n",
    "With the change of variables $\\bar y = \\ln y$ and $\\bar x_1 = \\ln x_1$, we can rewrite the above equation as:\n",
    "\n",
    "$$ \\bar y  = \\bar x_1 + x_2 t $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Check your answers:**\n",
    "\n",
    "We need to write the design matrix ${\\bf A}$ and the right-hand side vector ${\\bf \\bar y}$ needed to obtain the least squares solution ${\\bf \\bar x} = [\\bar x_1, x_2]$.\n",
    "\n",
    "Store the value of ${\\bf A}$ in variable `A_exp` and the value of ${\\bf \\bar y}$ in variable `ybar`.\n",
    "\n",
    "*Hint: Recall that [numpy.log](https://numpy.org/doc/stable/reference/generated/numpy.log.html) computes the natural logarithm by default.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "A_exp = np.array([np.ones_like(t_exp),t_exp]).T\n",
    "ybar = np.log(y_exp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, solve for the coefficients ${\\bf \\bar x}$ using the `least_sq` function you have defined above.\n",
    "\n",
    "Store the value of it in variable `xbar`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.85803595, 0.53002294])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xbar = least_sq(A_exp, ybar)\n",
    "xbar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Check your answers:**\n",
    "\n",
    "You can now use the change of variables to determine the coefficients $x_1$ and $x_2$ from the equation $y = x_1\\,e^{x_2 \\, t}$\n",
    "\n",
    "Store the value of them in variables `x1_exp` and `x2_exp` respectively.\n",
    "\n",
    "*Hint: You will need to use [numpy.exp](https://numpy.org/doc/stable/reference/generated/numpy.exp.html).*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "x1_exp = np.e**xbar[0]\n",
    "x2_exp = xbar[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the points and the fit together:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6b1ffbaa30>]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t_exp, y_exp, 'o')\n",
    "plt.plot(t_exp, x1_exp * np.exp(t_exp * x2_exp))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using real data: climate change"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Ice Extent](https://masie_web.apps.nsidc.org/pub/DATASETS/NOAA/G02135/north/monthly/images/10_Oct/N_198010_extn_v3.0.png)\n",
    "![Ice Extent](https://masie_web.apps.nsidc.org/pub/DATASETS/NOAA/G02135/north/monthly/images/10_Oct/N_201210_extn_v3.0.png)\n",
    "\n",
    "In addition to fitting functions to data points, we can use least squares to make predictions about future events.  In this example, we have a dataset containing the extent of arctic sea ice for each year since 1979 (given as the average during each October). We will fit least squares models to some of this data, using them to predict the extent of arctic sea ice in future years.\n",
    "\n",
    "*This is based on [\"Using Data from Climate Science to Teach Introductory Statistics\"](http://jse.amstat.org/v21n1/witt.pdf) and uses [NOAA data](https://nsidc.org/data/g02135) contributed by [Florence Fetterer](https://nsidc.org/research/bios/fetterer.html), et al.* "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>year</th>\n",
       "      <th>mo</th>\n",
       "      <th>data-type</th>\n",
       "      <th>region</th>\n",
       "      <th>extent</th>\n",
       "      <th>area</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>1979</td>\n",
       "      <td>10</td>\n",
       "      <td>Goddard</td>\n",
       "      <td>N</td>\n",
       "      <td>8.75</td>\n",
       "      <td>6.19</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1980</td>\n",
       "      <td>10</td>\n",
       "      <td>Goddard</td>\n",
       "      <td>N</td>\n",
       "      <td>9.18</td>\n",
       "      <td>6.50</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1981</td>\n",
       "      <td>10</td>\n",
       "      <td>Goddard</td>\n",
       "      <td>N</td>\n",
       "      <td>8.86</td>\n",
       "      <td>6.27</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1982</td>\n",
       "      <td>10</td>\n",
       "      <td>Goddard</td>\n",
       "      <td>N</td>\n",
       "      <td>9.42</td>\n",
       "      <td>6.67</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1983</td>\n",
       "      <td>10</td>\n",
       "      <td>Goddard</td>\n",
       "      <td>N</td>\n",
       "      <td>9.33</td>\n",
       "      <td>6.79</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   year   mo      data-type   region   extent     area\n",
       "0  1979   10        Goddard        N     8.75     6.19\n",
       "1  1980   10        Goddard        N     9.18     6.50\n",
       "2  1981   10        Goddard        N     8.86     6.27\n",
       "3  1982   10        Goddard        N     9.42     6.67\n",
       "4  1983   10        Goddard        N     9.33     6.79"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "ice_data = pd.read_csv(\"N_10_extent_v3.0.csv\", dtype={'year': np.int64, 'extent': np.double})\n",
    "ice_data.head() # print out first 5 rows of the data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have imported a data set for you using [Pandas](https://pandas.pydata.org/docs/user_guide/index.html), a library that provides routines for manipulating data tables.  *You do not need to know how to use Pandas for this activity or course*. Generally, Pandas is useful for organizing large, heterogeneous data sets (for which NumPy falls short)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1979, 1980, 1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989,\n",
       "       1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,\n",
       "       2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011,\n",
       "       2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "year = ice_data['year'].values\n",
    "year"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([8.75, 9.18, 8.86, 9.42, 9.33, 8.56, 8.55, 9.48, 9.05, 9.13, 8.83,\n",
       "       8.48, 8.54, 9.32, 8.79, 8.92, 7.83, 9.16, 8.34, 8.45, 8.6 , 8.38,\n",
       "       8.3 , 8.16, 7.85, 7.93, 7.35, 7.54, 6.04, 7.35, 6.92, 6.98, 6.46,\n",
       "       5.89, 7.45, 7.23, 6.97, 6.08, 6.77, 6.13, 5.73, 5.33, 6.77])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "extent = ice_data[' extent'].values\n",
    "extent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the first 22 years of the ice data to fit a linear function, and then predict future values of ice extent.  Below, we plot the years 1979 to 2000 with the corresponding ice extents."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6b1e392700>]"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_ice = 22\n",
    "\n",
    "plt.xlabel(\"Year\")\n",
    "plt.ylabel(\"Arctic Sea Ice Extent (1,000,000 sq km)\")\n",
    "plt.plot(year[:n_ice], extent[:n_ice], 'o')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Check your answers:**\n",
    "\n",
    "Using your `least_sq` function and general knowledge of least squares, fit a linear model to the ice extent:\n",
    "\n",
    "$$y = x_1 \\,t + x_2$$\n",
    "\n",
    "Store your coefficients in the array `x_ice_linear`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "t_ice = year[:n_ice]    # definition of t\n",
    "y_ice = extent[:n_ice]  # definition of y\n",
    "# Determine x_ice_linear using your least_sq() function\n",
    "A_ice = np.array([t_ice,np.ones_like(t_ice)]).T\n",
    "x_ice_linear = least_sq(A_ice, y_ice)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below is the plot of the 22 given points with those corresponding from the least squares model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6b1fff4e80>]"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "line_model = x_ice_linear[0]*t_ice + x_ice_linear[1]\n",
    "\n",
    "plt.xlabel(\"Year\")\n",
    "plt.ylabel(\"Arctic Sea Ice Extent (1,000,000 sq km)\")\n",
    "plt.plot(year[:n_ice], extent[:n_ice], 'o')\n",
    "plt.plot(t_ice, line_model , 's')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we want to use the same model (linear least squares fitted to years 1979 to 2000) to predict the ice extent beyond year 2000. For example, let's consider data from years 1979 to 2021:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1979., 1980., 1981., 1982., 1983., 1984., 1985., 1986., 1987.,\n",
       "       1988., 1989., 1990., 1991., 1992., 1993., 1994., 1995., 1996.,\n",
       "       1997., 1998., 1999., 2000., 2001., 2002., 2003., 2004., 2005.,\n",
       "       2006., 2007., 2008., 2009., 2010., 2011., 2012., 2013., 2014.,\n",
       "       2015., 2016., 2017., 2018., 2019., 2020., 2021.])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "years_plus = np.linspace(1979, 2021, 43)\n",
    "years_plus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Check your answers:**\n",
    "\n",
    "Determine the predicted ice extent values for the years 1979 to 2021 according to your `x_ice_linear` model from above.\n",
    "\n",
    "Store these values in the array `ice_plus`.\n",
    "\n",
    "*Hint: Do not recompute the model using the additional ice extent data. We can conceive the data from years 1979 to 2000 as our training set and the data from years 2001 to 2021 as our testing/validation set. You can click [here]( https://towardsdatascience.com/training-vs-testing-vs-validation-sets-a44bed52a0e1) learn more about training set vs. testing set vs. validation set in data analysis.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "ice_plus = np.array([years_plus,np.ones_like(years_plus)]).T @ x_ice_linear"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below, we plot each the actual and predicted data of ice extent for the years 1979 to 2021."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6b1e2f35e0>]"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xlabel(\"Year\")\n",
    "plt.ylabel(\"Arctic Sea Ice Extent (1,000,000 sq km)\")\n",
    "plt.plot(year, extent, 'o')\n",
    "plt.plot(years_plus, ice_plus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How does the predicted ice extent differ from the actual ice extent for each year?\n",
    "\n",
    "What is the estimated ice extent in 2021? What was the actual value?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.847075098814301"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ice_plus[-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.77"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "extent[-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The linear model does well until year 2000 (i.e. on the training set), then it fails to account for an apparent increase in the rate of ice melt. Let's use _quadratic_ model to try improving this:\n",
    "\n",
    "$$ y = x_1 \\, t^2 + x_2 \\,t + x_3 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Check your answers:**\n",
    "\n",
    "Using the same `y_ice` and `t_ice` as before (i.e. the training set of years 1979 to 2000), fit the quadratic model above to the ice extent (that is, solve for $x_1$, $x_2$, and $x_3$).\n",
    "\n",
    "Store your coefficients in the array `x_ice_quad`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "#grade (enter your code in this cell - DO NOT DELETE THIS LINE) \n",
    "A_ice2= np.array([t_ice**2,t_ice,np.ones_like(t_ice)]).T\n",
    "x_ice_quad = least_sq(A_ice2,y_ice)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We plot the quadratic model below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6b1e2c9700>]"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "quad_model = x_ice_quad[0]*years_plus**2 + x_ice_quad[1]*years_plus + x_ice_quad[2]\n",
    "\n",
    "plt.xlabel(\"Year\")\n",
    "plt.ylabel(\"Arctic Sea Ice Extent (1,000,000 sq km)\")\n",
    "plt.plot(year, extent, 'o')\n",
    "plt.plot(years_plus, quad_model )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This model better captures the ice extent beyond year 2000.\n",
    "\n",
    "\n",
    "**Try this:**\n",
    "\n",
    "In what year does this quadratic model predict that the ice extent will reach 0 (i.e. the artic ice will have completely melted)?\n",
    "\n",
    "Solve $x_1\\,t^2 + x_2\\,t + x_3 = 0$ with your group to compute the answer below.\n",
    "\n",
    "*Hint: You seek the solutions to a quadratic equation, the analytical formula for which you have likely memorized.*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
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